My question is does [tex]\rho[/tex] include the density contribution for Cosmological Constant (dark energy) [tex] \Lambda [/tex] or is this derivation only for a Universe with no cosmological constant?

How does one then actually measure the density of Universe?

I know that the density has been measured to be slightly less than the critical density, but I thought we are meant to live in a flat Universe? Is this due to the cosmological constant and how is this reconciled with [tex]\rho[/tex] not being exactly [tex]\rho_c[/tex]?

Finally, I am right in saying that a Universe with [tex]\rho_c[/tex] will stop expanding after infinite time, one with [tex]\rho > \rho_c[/tex] will collapse back on itself and one with [tex]\rho < \rho_c[/tex] will expand forever?

Yes, [itex]\rho[/itex] includes contribution from the cosmological constant. In other words, we can write the density as a function of scale factor as
[tex]
\rho = \rho_c\left(\Omega_Ma^{-3} + \Omega_Ra^{-4} + \Omega_{\Lambda}\right)
[/tex]

Finally, I am right in saying that a Universe with [tex]\rho_c[/tex] will stop expanding after infinite time, one with [tex]\rho > \rho_c[/tex] will collapse back on itself and one with [tex]\rho < \rho_c[/tex] will expand forever?

This is basically correct, although I don't think, "stop expanding after infinite time" is a well-defined notion.

How does one then actually measure the density of Universe?

Fit supernova data and/or CMB data to different models and see what works best.

I know that the density has been measured to be slightly less than the critical density, but I thought we are meant to live in a flat Universe? Is this due to the cosmological constant and how is this reconciled with [tex]\rho[/tex] not being exactly [tex]\rho_c[/tex]?

The measurement of [tex]\rho[/tex] is within error of being less than, equal to, or greater than the critical density. People say we live in a "flat universe", because the measured value is very close to the critical density.